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Example: (1.) To reduce pounds sterling to shillings, pence, and farthings. Since there are 20 shillings in each £, 12 pence in each s., and 4 farthings in each d., it must be evident that the number of £s. multiplied by 20 will give the equivalent number of s.; the number of s. multiplied by 12 will give the number of d.; and the number of d. by 4, the number of farthings. Thus,

£15

£9 78. 10d. 20

20

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144008. 9017f. Example : (2.) To reduce farthings to pence, shillings and pounds.-A process the reverse of that above is necessary in this case. Thus :

4)14400f.

4)9017

:

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15€

£9 78. 10d.
Example : (3.) To reduce cuts. to grs. and lbs.; and lbs. to grs. and cuts.
cwt.
cwt. qrs. lbs.

lbs.
Example: 9
11 2 17

7)1305
4
4

28

4)186—3

36 qrs.

46 qrs.

28

28

4)46—17

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1008 lbs. 1305 lbs. The following expeditious methods of reducing cwts., qrs. and lbs. to total lbs., will be found useful when a table is not at hand, by officers surveying distillers and rectifiers.

Reduce the qrs. and lbs. to total lbs. mentally, reckoning 28 lbs. for 1 gr., 56 for 2, fc. : then, 1st multiply the cwt. by 12, not writing down the multiplier, and add in the lbs. as the work proceeds ; set the product under the cwt., two places to the right, and add it to the cut. or 2nd, multiply the cut. by 2, add in two figures to the right and also the pounds, as the work proceeds. Examples: (1.) 1st method.

2nd method. cwt. qrs. lbs.

cwt. qrs. lbs. 9 11

9 3 11

3

9 95
1103 lbs.

(x by 12)

9 95
1103 lbs.

(x by 2)

Ex. (2.)

14 1 25

14 1 25

14 53

14 53 (x 12) 1621 lbs.

(x2) 1621 lbs. In the first of these methods, the cwts. are supposed to be multiplied by 100, to which is added the product of the cwt. by 12, increased by the lbs. from the given qrs. and lbs.

In the second method, the cwts. are multiplied by 10 and 11 successively, by supposing a cipher annexed, and adding in two figures to the right; to this is added the product of the cwt. by 2, increased by the lbs. from the given qrs. and lbs.

To reduce Quarters, Bushels and Gallons to Gallons, and the reverse.

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To reduce Quarters and Bushels of Malt and Lbs. of Sugar, to an equivalent in Barrels of Beer.

A brewer is deemed by law to have brewed one barrel of beer for every two bushels of malt, and the same for every fifty lbs. of sugar consumed by him. Malt.

Sugar. qrs. bus.

lbs. Example: 615 7

895 4

-02

2463.5 barrels.

17.90 barrels The 7 bushels of malt are equal to 3-5 barrels ; and to reduce the quarters to barrels we multiply by 4 as the shortest way of multiplying by 8 and then dividing by 2. In reducing the lbs. of sugar, multiplication by .02 is substituted for division by 50, since zo=:02

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To reduce Barrels, Firkins, and Gallons to Gallons, and the reverse.

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If there are 64, 6, and 64 bottles respectively to the gallon, how many gallons are represented in each case by 230 bottles ?

230

2

230

3

230

4

2

5

13) 460(351
19) 690(3620

25)920(364
70
120

170
5
6

20 Answers. 35 galls. 36 galls. 364 galls.

To divide by 64 or ys is to multiply by 1s, that is, to multiply by 2 and divido by 13. Similarly, to divide by 61 is to multiply by *, and to divide by 61 is to multiply by an A shorter way of obtaining the last result would be to multiply by •16 (=i=106)

Reduction of Spirits to Proof, 8c.—For the present purpose it will be sufficient to explain, that the strength or alcoholic value of commercial spirits in this country is denoted by a number which expresses its relation to a standard strength, called, Proof, and represented numerically by 100. Proof or 100 is supposed to be divided into 100 equal parts or degrees, and each of these again into tenths.

A spirit stronger than Proof is said to be so many degrees per cent. over Proof (O. P.) and a spirit weaker than proof, to be so many degrees under proof (U.P.) the term per cent. being here employed, not in its proper arithmetical sense, but merely to refer the expression of strength to 100.* For example, a spirit at « 25 per cent. O.P.” is understood to be once and twenty-five hundredths or once and a quarter as strong as proof; it contains as much alcohol as proof spirit contains and 25 per cent. more. Similarly, “ 25 per cent. U.P.” means that a spirit of that strength wants 25 parts in 100 of being as strong as proof, or that 75 parts out of the 100 consist of proof spirit and the remainder of water. In valuing spirits, it is always assumed that the mixture is composed of water and alcohol only. The proof value of any spirit of given strength is expressed, therefore, by adding degrees 0. P. to 100, and subtracting degrees U. P. from 100: thus, the value of a spirit at 60 O.P. is represented by 160 (=100+60) and the value of a spirit at 60 U.P. by 40 (=100—60.)

In the former case, 100 gallons of the spirit are equal in value to 160 gallons, and in the latter case, to only 40 gallons at proof strength. But if we divide the value of 100 gallons in any instance by 100, we shall obtain a factor or multiplier which will reduce any quantity of spirits to an equivalent at proof. As 100 gallons at 60 0. P. are equal in value to 160 at proof, 1 gallon is equal to 1.60 at proof, and so on. Hence, a general rule for reducing spirits of a given strength to proof. RULE.--Add degrees 0. P. to 100: and subtract degrees U. P. from 100 : divide the sum or difference by 100, and multiply the given quantity by the number so obtained.

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* On this point see the "Handbook for Officers of Exciso," pages 115, 116

Example (1.) Reduce to proof 170 gallons of spirits at 12.5, 35.7 and 54:0 per cent O. P. respectively. Degrees O. P. 12.5

35.7

54
Add
100
100

100

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Example (2.) Reduce to proof 93 gallons at 16:9, 21.4, and 80-0 per cent U. P. respectively.

100
100

100
Subtract degrees U. P. 16.9

21.4

80

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As it is not the practice in revenue calculations to retain any decimals beyond tenths of a gallon at proof, most questions in the reduction of spirits may be wrought with sufficient exactness by the method of contracted multiplication. (See page 85.)

Spirits may also be reduced to proof in the following manner : multiply the quantity by the rate per cent 0. P. or U. P. divided by 100. If the given strength be 0. P., add this product to the quantity ; if U. P., subtract it from the quantity, thus

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The first method is that universally employed by the Excise as the most simple and direct. In the Customs department, the second mode of calculation is usually resorted to, as it exhibits what is called the “overproof” or “underproof” of a quantity of spirits,* and it is the practice of the Customs in some cases to enter this “overproof,” &c., separately, in their accounts and documents. Each gallon at 12.5 0. P., exceeds in value a gallon at proof by •125 of a proof gallon, and the “overproof,” or excess beyond proof strength of 170 gallons, is, as above shown, 21•25 proof gallons ; to which, if there be added the bulk gallons-estimated as of proof strength-we have for the total equivalent at proof 191-2 gallons. Similarly, with respect to the strength U. P., each gallon at 16.9 U. P. falls short of the value of a gallon at proof by •169 of a proof gallon. 93 gallons are therefore in defect by 15717 proof gallons, and this amount of

underproof” subtracted from the bulk gallons estimated as of proof strength, leaves 77.2 proof gallons.

In the computation of proof quantities, the rule of the Customs is the same as that of the Excise—to retain only the first place of decimals or tenths of a proof gallon ; but when the system of finding the amount of “overproof” or « underproof” is adopted, instead of the process used by the Excise, it is necessary in the case of underproof spirits, to deduct two decimal places of the amount of “underproof” from the bulk quantity, so that the trader may not be charged more than he would be if the proof gallons were otherwise computed. Thus, let the amount of “underproof” be 15.717 gallons, and the bulk gallons 93, as before. If 15.7 gallons only were deducted, the charge would be 77.3 proof gallons, but if 15.71 be deducted, the charge will be 77.2, the same result as that obtained by subtracting the rate per cent. U. P. from 100, and employing one hundredth of the difference as a factor, in the way exemplified on page 99.

When spirits are taken account of in bottles, the quantity is reckoned by the Customs in gallons and gills, that is, 32nds of a gallon, and in casting out the strength of such spirits, if the first decimal figure be •5 or upwards it is charged as a gill, if less than 5 it is disregarded. For instance, 163. gallons at 8.4 per cent. U. P. would be reduced to proof as follows—

1637 gallons = 540 gills.

·084 factor for strength.

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15 Gallons at proof. To reduce spirits at a given strength to any other strength per cent O. P. or U. P. we may employ this GENERAL RULE.--Multiply the sum or difference of the GIVEN rate per cent. and 100, by the quantity of spirils, and divide the product, by the sum or difference of the REQUIRED rate per cent. and 100.

* The “Overproof” as determined in this manner represents also the quantity of water neces sary for reducing the given number of gallons of spirits to proof strength. (See page 102 )

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